Structures built with trusses consisting of tetrahedrons and/or octahedrons have been in use for many years. They can be seen today in shopping malls and other buildings as roofing trusses, as dividing walls, as framework for the support of light fixtures, sprinkler systems, ceiling panels, advertising displays, etc. One of the most inspiring representative of such structures is the Olympic Stadium in Berlin, West Germany, which has a contilevered space frame roof structure of this kind, which was built by Unistrut Building Systems of Wayne, Mich., USA. A U.S. Pat. No. 2,986,241, issued to Mr. R. B. Fuller, provides a comprehensive description of octahedral-tetrahedral trusses. Emphasis is given to the highly favorable weight to strength ratio inherent to truss designs of this type. Mr. Fuller's patent prescribes the exclusive use of equilateral triangles with which a wide selection of structural trusses for wall, roof and floor designs, related to the rectangular prism rather than spherical space enclosures, can be built. Another patent, also issued to Mr. Fuller, U.S. Pat. No. 3,354,591, describes an octahedral tensegrity system, being a flexible truss. It was used for the construction of the United States Pavilion in Montreal, Quebec, Canada at "EXPO '67". The structure is geodesic of which octahedrons are connected to each other. Each octahedron is created by having its X, Y and Z axes represented by three rigid rods as compression members. The six rod ends are held in position by tension wires. The outer surface, according to the drawing for his patent, consists of small hexagonal and pentagonal pyramids which are separated by small recessed triangular panels. The Montreal structure had additional connecting rods between all pyramidal vertices installed. A substantial increase in structural strength was achieved with this addition, however, the result was also increased complexity and higher construction cost.
While studying geometric solids and constructing models, I discovered one type of isosceles triangle of specific angular values and proportions, which possesses extraordinary features: (a) It can be described as a "divine" or "golden" triangle, because the golden ratio is evident on its face many times, when lines are drawn from any of its three vertices across the triangle to intersect at right angles the line opposite of the vertex. The golden ratio is always found in the proportion of at least one part of the divided line to the line drawn; (b) In its plurality, this isosceles triangle, used in the assembly of tetrahedrons and octahedrons, which in turn are assembled into specific patterns--explained later in this text--produces three distinctly different forms of space filling and close-packing of geometric solids, which produces a simultaneous spherical super-symmetry. (c) The combination of the three forms of close-packing of the above mentioned octahedrons and tetrahedrons permits the construction of the framework of icosahedral-spherical space-enclosing super-structures with the exclusive use of the said isosceles triangle.
Close-packing features of geometric solids have fascinated scientists and mathematicians for millennia. One reason for this continued interest may be the need for simpler, lighter, stronger and less expensive construction methods for buildings and space enclosures of all kinds, involving less material while employing the least possible number of types of identical structural elements. My tetrahedral-octahedral truss design fulfills the aforesaid need on a broad scale, and it is especially suited for spherical and dome-like structures.
The three distinctly different types of close-packing of geometric solids are three-axial, six-axial and ten-axial. All three are present simultaneously in the truss system described in this invention.
Close-packing is the feature of some geometric solids such as cubes, tetrahedrons, octahedrons etc., which allows them, in their plurality either alone or in combination with others, to be attached to each other without leaving any space between them.
Three-axial pertains to the intersecting of three straight lines (axes) at one common point. This configuration can also be visualized as six lines (vectors) radiating outward from that one common point in space, in a specific symmetrical way.
Three-axial close-packing is the feature of certain polyhedrons, of which space-filling occurs outward in six distinctly different linear directions, as polyhedrons are added to the assembly. The type of polyhedron of the system described in this invention has six sides. The sides are congruent rhombi, and the polyhedron is a rhombic hexahedron, therefore, the close-packing feature is rhombic-hexahedral, however, each rhombic hexahedron consists of one octahedron as FIG. 8a two tetrahedrons as FIG. 4.
Three-axial close-packing pertaining to this invention begins at one point in space, from which tetrahedrons and octahedrons can be attached to each other in the direction of the six vectors. It applies to the construction of linear and areal trusses, not intended as part of any claims of this patent, for roofs, walls and floors of various shapes. Surface areas of areal trusses may have many shapes. In their triangular surface form, however, they are used as sectional building blocks for all icosahedral spherical structures of this invention. The building blocks are called "truss components".
Six-axial close-packing is stellar-dodecahedral, which is intrinsic to the regular pentagonal dodecahedron. It has six axes intersecting at one point in space, which can also be considered as twelve vectors radiating outward from one central point. Close-packing of tetrahedrons and octahedrons occurs, when tetrahedrons and octahedrons are grouped in five or multiples of five in pentagon form around the twelve vectors in alternating succession, octahedrons onto tetrahedrons and tetrahedrons onto octahedrons. The vector lines (0 to point 36-n, FIG. 30) radiate outward from the central point "O", through the center points of the twelve pentagon faces of a pentagonal dodecahedron. Six-axial close-packing can clearly be recognized in the structural truss development of spherical structures related to the pentagonal dodecahedron.
Ten-axial close-packing allows the construction of spherical structures based on the regular icosahedron. Ten axes are intersecting at one point in space being the geometric center of an iscosahedron. This configuration must be visualized as twenty lines radiating outward in complete symmetry from the central point "O", through the centers of the twenty planar equilateral triangle faces of the icosahedron. The lines are vectors, in the outward directions of which ten-axial close-packing of tetrahedral and octahedral elements takes place.
My truss system in both, areal and icosahedral form has many uses, especially in outer space, where the basic tetrahedral and octahedral building elements could be employed for the construction of planar structures, such as platforms or extensive reflective areas and also for spherical-icosahedral space enclosures of many sizes, from small satelites to gigantic stations. On earth, my truss system is also very useful for dome-like structures, such as IMAX movie theatres, homes, arctic dwellings and shelters of many kinds, for storage containers, playground equipment such as climbers or supports for slides etc., for educational building sets and for the construction of artistic geometric sculptures. The fact, that only one type of isosceles triangle, in its plurality as sheet, panel or strut-arrangement is required for the construction of all of the above applications, would greatly simplify the process of manufacturing and construction, in comparison to present available processes.